Our primary goal is to find, among the convex polygons inscribed in a given convex \(n\) -gon, one with
minimum area or perimeter. For these problems we give algorithmic solutions requiring \(O(n)\)  and \(O(n^3)\)  steps, respectively.
We will see that the first problem is relatively easy to solve, but the second one is not. This problem can be regarded as a variant of the problem of finding the shortest closed billiard trajectories in a given convex polygon. We note that this problem, proposed also for convex bodies in general, is an extensively studied area of research closely related, among other things, to dynamical systems and symplectic geometries. For more information on this subject, the reader is referred to the papers [1]}, [2]}, [3]}, [4]}, the book [5]}, or the video recording of the highly interesting talk [6]} of Artstein-Avidan.
